VerifiedHint: We will first draw a figure of line y = mx, y = mx + 1 with a positive slope. Then, we will draw the lines y = nx and y = nx + 1 with a negative slope. We are taking opposite signs for the convenience of the figure and it is not a compulsory condition. From the figure, we will see that two triangles are formed with the y – axis. The area of the parallelogram is twice that of the two triangles formed.
Complete step-by-step answer:
The figure of the parallelogram will be as follows:
The point of intersection of y = mx and y = nx + 1 is B, y = mx + 1 and y = nx + 1 is A and intersection of y = mx + 1 and y = nx is C. So, A, B, C and O form a parallelogram.
From the figure we can see that two triangles OAB and OAC are formed with OA as the common base.
OB = CA and AB = OC, since they are opposite sides of a parallelogram.
Thus, we can say that triangle OAB and triangle OAC are congruent triangles.
Therefore, the area of the parallelogram is twice the area of one of the triangles.
Area of a triangle is half the product of base and height of a triangle.
the distance of B from y – axis will be the height of the triangle OAB
We will now find point B.
To find point B, we need to solve y = mx and y = nx + 1
$ \Rightarrow $ mx = nx + 1
$ \Rightarrow $ (m – n)x = 1
$ \Rightarrow $ x = $ \dfrac{1}{\left( m-n \right)} $
Thus, the height of triangle OAB is $ \dfrac{1}{\left( m-n \right)} $ .
Now, we will find the length of the base OA.
A lies on the y – axis, thus x – coordinate of A is 0.
Therefore, substitute x = 0 in y = mx + 1.
$ \Rightarrow $ y = m(0) + 1
$ \Rightarrow $ y = 1
Thus, the length of OA is 1 unit.
$ \Rightarrow $ Area of parallelogram = 2(Area of triangle OAB)
$ \Rightarrow $ Area of parallelogram = $ 2\times \dfrac{1}{2}\times 1\times \dfrac{1}{\left( m-n \right)} $
$ \Rightarrow $ Area of parallelogram = $ \dfrac{1}{\left| m-n \right|} $
The modulus is applied as the area cannot be negative.
So, the correct answer is “Option D”.
Note: Another method to solve this will be through option verification method. Students can take any arbitrary form m and n and then find the area of parallelogram with real numbers. While taking values of m and n, students are advised to check each option and make sure each option is yielding different values.
